that is, \({\mathcal {D}}_W\)  is the set of all entangled states that
detected by \(W\) . For entanglement witnesses \(W_1,W_2\) , we say that
\(W_1\)  is finer than \(W_2\)  if \(\mathcal {D}_{W_2}\subset \mathcal {D}_{W_1}\) , denoted by \(W_2\prec W_1\) . While, an
entanglement witness \(W\)  is optimal if there exists no other
witness finer than it. Obviously, a state \(\rho \)  is entangled if and
only if there is some optimal EW such that \({\rm Tr}(W\rho )<0\) . In
[1]}, Lewenstein, Kraus, Cirac and Horodecki proved that: (1)
\(W\)  is an optimal entanglement witness if and only if \(W-Q\)  is no
longer an entanglement witness for arbitrary positive operator \(Q\) ;
(2) \(W\)  is optimal if \({\mathcal {P}}_W= \lbrace |e,f\rangle \in H\otimes K:\langle e, f|W|e,f\rangle =0\rbrace \)  spans the whole \(H\otimes K\)  (in this
case, we say that \(W\)  has spanning property). For the infinite
dimensional version of these results, see [2]}. To the best of
the author's knowledge, the above criterion (2) is the only method
we have known by now that is practical of checking optimality of
witnesses. In fact, almost all known optimal EWs are checked by
using of the criterion (2) (Ref. [3]}, [4]} and the references
therein). However, the criterion is only a sufficient condition.
There are known optimal witnesses that have no spanning property.
For example, consider the Choi map \(\phi \)  from \(M_3\)  into \(M_3\) 
defined by
\( \left(\begin{array}{ccc} a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\ a_{31}&a_{32}&a_{33}\end{array}\right)\mapsto \left(\begin{array}{ccc} a_{11}+a_{33}&-a_{12}&-a_{13}\\-a_{21}&a_{22}+a_{11}&-a_{23}\\-a_{31}&-a_{32}&a_{33}+a_{22}\end{array}\right). \qquad \mathrm {(0.1)}\) 
